Lacunary growth implication for Problem 995

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This exposition is Plasma AI's. The source construction is Ho's; the claim page holds the independent whole-proof examination and separate PASS grade for the $p=2$ specialization and the little-o consequence written below.

Lacunary sums

The retained claim of Ho's Theorem 1.6 states that at $p=2$ there are a mean-zero $f\in L^2(\mathbb T)$ and an integer sequence with $n_{j+1}/n_j\ge2$ such that, for almost every $x$,

$$\limsup_{N\to\infty} \frac{\sum_{j\le N}f(n_jx)}{N(\log N)^{1/2-\varepsilon}} =+\infty \qquad\text{for every }\varepsilon>0.$$

Use only $\varepsilon=1/4$. The ratio

$$\frac{N(\log N)^{1/4}}{N\sqrt{\log\log N}} =\frac{(\log N)^{1/4}}{\sqrt{\log\log N}} \longrightarrow+\infty.$$

On the almost-everywhere set given by that one value of $\varepsilon$, the positive limsup therefore excludes $\sum_{j\le N}f(n_jx)=o(N\sqrt{\log\log N})$. Using a periodic representative identifies $f(n_jx)$ with $f(\{n_jx\})$; changing the representative on a null set affects only a countable union of null preimages. The factor-two lacunarity and mean-zero condition are at least as strong as the catalog hypotheses and match the additional centering condition in the formal-conjectures statement.

Thus the displayed little-o question in Problem 995 has a negative answer. The filed review and grade cover both the source's $p=2$ construction and this implication, with the exact foundational and source-fidelity qualifications recorded on the claim page. The broader all-$p$ assertions, Remark 7.1 and the growth of each individual function and sequence are outside that acceptance.